KPS Conjecture & Escape of Mass

• KPS Conjecture is a theory in Diophantine approximation that predicts full escape of mass in the continued fraction expansions of quadratic Laurent series over finite fields.
• Counterexamples like the p-Cantor sequence and Thue–Morse case show that the full escape property fails, motivating refined notions of maximal and generic escape.
• Key techniques such as number walls and Toeplitz determinants reveal a geometric-fractal structure connecting automatic sequences to homogeneous dynamics.

The term "KPS Conjecture" typically refers to a conjecture proposed by Kemarsky, Paulin, and Shapira in the field of Diophantine approximation over function fields, specifically concerning the "escape of mass" phenomenon for quadratic irrationals described by Laurent series over finite fields. This conjecture, and more recent refinements and counterexamples, has catalyzed new directions in the arithmetic and dynamics of positive characteristic, linking continued fraction expansions to homogeneous dynamics and the theory of automatic sequences.

1. Statement and Motivation

The original KPS conjecture posits a universal "full escape of mass" property for Laurent series with coefficients in $\mathbb{F}_p$, the finite field with $p$ elements, under scaling by powers of irreducible polynomials. Let $\Theta(t)\in\mathbb{F}_p((t^{-1}))$ be a quadratic irrational (i.e., a nonperiodic Laurent series satisfying a quadratic relation), and let $P(t)\in\mathbb{F}_p[t]$ be an irreducible polynomial. The conjecture asserts:

For every such pair, as one considers the family $\{P(t)^k\Theta(t)\}_k$, the continued fraction expansion of each $P(t)^k\Theta(t)$ (which can be written as $[A_0;A_1,\ldots,A_{\ell(k)}]$) should, in the limit, exhibit "full escape of mass" in the following quantitative sense:

$$e_k = \lim_{n \to \infty} \liminf_{k \to \infty} \frac{\sum_{i=1}^{\ell(k)} \max\{\deg(A_i) - n, 0\}}{\sum_{i=1}^{\ell(k)} \deg(A_i)} = 1$$

This predicts that, eventually, the "mass" (as measured by degrees of the partial quotients) concentrates in arbitrarily large-degree terms, reflecting a kind of ergodic "escape" to infinity analogous to measure-theoretic escape of mass in homogeneous dynamics.

2. Mathematical Framework and Objects

A precise formulation of the escape ratio utilizes the continued fraction expansion in the Laurent series field $\mathbb{F}_p((t^{-1}))$. For

$\Theta(t)= t^{-1}\sum_{i=0}^\infty s_i t^{-i}$

and its expansion

$\Theta(t) = [A_0(t); A_1(t), A_2(t), \ldots]$

define for each $k$, after applying $P(t)^k$ and extracting the continued fraction, the sums: $$e_{k,n}(\Theta) := \frac{\displaystyle \sum_{i=1}^{\ell_k}\max\{(h_{k,i}-h_{k,i-1}-n),0\}}{\displaystyle h_{k,\ell_k}-h_{k,0}}$$ where the $h_{k,i}$ index the rows of zeros in the number wall representation of the associated sequence (see below). The conjecture requires $e_k\to1$ as $k,n\to\infty$.

The central algebraic objects are Laurent series in $\mathbb{F}_p((t^{-1}))$ and their continued fraction expansions under multiplication by polynomials. The associated "number wall" is a two-dimensional array $W(S)[m,n]$ of Toeplitz determinants formed from the coefficient sequence $S=(s_i)$, where $W(S)[m,n]=\det(T_S(n;m))$ encodes combinatorial information about runs of zeros and, indirectly, the degrees of the partial quotients $A_i(t)$.

3. Counterexamples and Refined Notions

The paper (Aranov et al., 22 Oct 2025) rigorously disproves the generality of the KPS conjecture by constructing explicit counterexamples. For $p=2$ and $P(t)=t$, prior work showed that the Thue–Morse sequence (with its associated Laurent series) yields an escape ratio strictly less than $1$, precisely $2/3$. The present work generalizes this to all odd primes $p$:

• For the $p$-Cantor sequence—a $p$-automatic sequence constructed via a specific $p$-morphism—the associated Laurent series $\Theta_p(t)$ satisfies

$$\lim_{n\to\infty}\lim_{k\to\infty} e_{k,n}(\Theta_p) = \frac{2}{p}$$

implying that the "full escape" property fails in these cases as well.

To account for this, the authors introduce two refinements:

• Maximal Escape of Mass: Replacing $\liminf$ with $\limsup$ in the above limits, they define the maximal escape parameter. Full maximal escape means achieving a value of $1$ along a subsequence of the indices $k$.
• Generic Escape of Mass: If, for any $\varepsilon>0$, the set of indices $k$ with $e_{k,n}(\Theta)>1-\varepsilon$ has density $1$, then the sequence exhibits full generic escape. Both maximal and generic escape were shown to hold in all previously known counterexamples, suggesting they capture more typical behavior.

4. Key Technical Constructions

A central technical device in (Aranov et al., 22 Oct 2025, Robertson et al., 22 Oct 2025) is the use of number walls. Given $S=(s_i)$, the "number wall" $W(S)[m,n]$ is defined as the determinant of the Toeplitz matrix $T_S(n;m)$: $$T_S(n;m) = \begin{pmatrix} s_n & s_{n+1} & \cdots & s_{n+m} \ s_{n-1} & s_n & \cdots & s_{n+m-1} \ \vdots & \vdots & \ddots & \vdots \ s_{n-m} & s_{n-m+1} & \cdots & s_n \end{pmatrix}$$ Each zero run in the number wall directly corresponds to the degree of a partial quotient. The "window structure theorem" specifies that zeros in the wall appear in square windows, and this recursive structure is described by a set of frame equations (FC1, FC2, FC3).

For the $p$-Cantor sequence, these structure theorems yield an explicit calculation of the minimal escape ratio and establish the automaticity and periodicity properties needed for the refined analysis. Furthermore, the authors develop a geometric connection: by coding the nonzero entries of the number wall into a subset of $[0,1]^2$, a fractal of explicit Hausdorff dimension $\log\left(\frac{p^2+1}{2}\right)/\log(p)$ emerges (Robertson et al., 22 Oct 2025).

5. Implications, Broader Context, and Future Directions

The disproof of the original KPS conjecture demonstrates that the dynamics of escape of mass for Laurent series in positive characteristic are subtler than their real counterparts. Full escape (liminf = 1) is not universal; yet maximal and generic escape may be robust phenomena. These insights refine expectations for Diophantine approximation, continued fraction expansions, and dynamics on spaces like $$\mathrm{SL}_2(\mathbb{F}_p((t^{-1})))/\mathrm{SL}_2(\mathbb{F}_p[[t]])$$.

Open problems and directions include:

• Classification: Determining for which quadratic irrationals or which subclasses of automatic sequences full, maximal, or generic escape occurs.
• Dynamical significance: Understanding how escape of mass corresponds to measure-theoretic entropy, orbit closures, or spectral properties of associated flows.
• Extension to higher rank and multidimensional settings: Adapting "number wall" techniques and automatic sequence analyses to more general algebraic groups, fields, or multidimensional expansions.
• Fractal invariants: Further analysis of the fractal sets arising from the number wall, with connections to discrepancy, spectral theory, and combinatorics.

6. Table: Key Concepts and Quantities

Concept	Definition/Formula	Role
Escape ratio $e_{k,n}$	$$\frac{\sum_{i=1}^{\ell_k}\max\{(h_{k,i}-h_{k,i-1}-n),0\}}{h_{k,\ell_k}-h_{k,0}}$$	Measures escape of mass
Full escape	$\lim_{n\to\infty} \liminf_{k\to\infty} e_{k,n}=1$	KPS conjecture's prediction
Maximal escape	$\lim_{n\to\infty} \limsup_{k\to\infty} e_{k,n}=1$	Holds in the refined conjecture
Generic escape	Density-1 set of $k$ with $e_{k,n} >1-\varepsilon$	Captures typical behavior
Number wall $W(S)[m,n]$	$\det(T_S(n;m))$	Encodes partition structure

7. Conclusion

The KPS conjecture, in its original "full escape" formulation, is falsified by the existence of counterexamples such as the $p$-Cantor sequence for all odd primes and the Thue–Morse case for $p=2$ (Aranov et al., 22 Oct 2025, Robertson et al., 22 Oct 2025). Nonetheless, maximal and generic forms of escape of mass appear to be generic in these and possibly all cases, suggesting refined conjectures better reflecting the arithmetic and dynamical realities of Laurent series over finite fields. The technical innovations—particularly the construction and analysis of number walls—provide a framework for future studies in combinatorial, dynamical, and fractal aspects of positive characteristic Diophantine approximation.